MAT 1236
Calculus III
Section 11.9
Representations of
Functions as Power Series
http://myhome.spu.edu/lauw
HW & Misc. Items
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WebAssign 11.9
Be sure to resolve all the HW problems
that you do not know how to do
Preview
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Remarks on Differentiation and Integration
of Power Series
Representation of functions of the form
1
1  g ( x)
as Power Series
Differentiation and Integration
of Power Series

If f ( x)   cn ( x  a) n
n 0

then f ( x)   ncn ( x  a) n 1
n 0
and

( x  a) n 1
f ( x)dx  C   cn
n 1
n 0

Theorem 2
Diff. and Integration of Power Series
Radius of
Convergence
I.OC.
f ( x )   cn ( x  a ) n
𝑅
𝐼1
f ( x)   ncn ( x  a ) n 1
𝑅
𝐼2
( x  a) n1
f ( x)dx  C   cn
n 1
n 0
𝑅
𝐼3

n 0

n 0


Summary
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We can perform term-by-term differentiation
and integration on power series.
The radius of convergence does not change
upon diff. or integration. But the I.O.C. may
be different at the end points.

f ( x)   ncn ( x  a)
n 0
n 1

  ncn ( x  a)
n 1
n 1
Summary



We can perform term-by-term differentiation
and integration on power series.
The radius of convergence does not change
upon diff. or integration. But the I.O.C. may
be different at the end points.

f ( x)   ncn ( x  a)
n 0
n 1

  ncn ( x  a)
n 1
n 1
Remarks

f ( x)   ncn ( x  a)
n 0
n 1

  ncn ( x  a )
n 1
n 1
Applications….
Series expansions of functions similar to
1
1  g ( x)
have many applications.
Consider the series

n
2
3
[
g
(
x
)]

1

g
(
x
)

[
g
(
x
)]

[
g
(
x
)]


n 0
This is a G.S. with 𝑎 = 1 and 𝑟 = 𝑔(𝑥)
The series is convergent if g ( x)  1
1
(the sum is
)
1  g ( x)
and divergent if
g ( x)  1
In other words…

1
  [ g ( x)]n
1  g ( x) n 0
The series is convergent if
and divergent if
g ( x)  1
g ( x)  1
In other words…

1
  [ g ( x)]n
1  g ( x) n 0
The series is convergent if
and divergent if
g ( x)  1
g ( x)  1
Attention: The criteria are similar to but
different from those of the Ratio Test
because they are based on the G.S.
Example 1
Represent the following function as power
series and find the I.O.C. and 𝑅
1
2
1 x
Example 1

1
  [ g ( x)]n
1  g ( x) n 0
Example 2
Represent the following function as power
series and find the I.O.C. and 𝑅
1
2
1 x
Example 2

1
  [ g ( x)]n
1  g ( x) n 0
Example 3
Represent the following function as power
series with center 𝑎 = 0 and find the I.O.C.
and 𝑅
1
3 x
Example 3

1
  [ g ( x)]n
1  g ( x) n 0
Example 4
Represent the following function as power
series with center 𝑎 = 0 and find the 𝑅 (No
I.O.C.).
1
2
(3  x)
Example 4
18-Point Decision Chart
Challenge
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The decision chart will be judged by

Must be software generated charts.
Deadline: 6/1 Monday at 5pm.
Must be original, do not copy from the
web!
(I already have 2 submissions!)
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• Accuracy and completeness
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